Cluster Variables on Certain Double Bruhat Cells of Type $(u,e)$ and Monomial Realizations of Crystal Bases of Type A

TL;DR

This paper explores the cluster algebra structure of double Bruhat cells in type A, linking generalized minors to monomial realizations of Demazure crystals, providing new insights into their algebraic and combinatorial properties.

Contribution

It introduces a novel description of generalized minors as sums of monomial realizations of Demazure crystals for specific double Bruhat cells in type A.

Findings

Generalized minors can be expressed as sums over Demazure crystal monomials.

The cluster variables relate to monomial realizations of crystal bases.

The approach bridges cluster algebra and crystal basis theories in type A.

Abstract

Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$, $B$ and $B_-$ be two opposite Borel subgroups in $G$ and $W$ be the Weyl group. For $u$, $v\in W$, it is known that the coordinate ring ${\mathbb C}[G^{u,v}]$ of the double Bruhat cell $G^{u,v}=BuB\cap B_-vB_-$ is isomorphic to an upper cluster algebra $\bar{{\mathcal A}}({\bf i})_{{\mathbb C}}$ and the generalized minors $\{\Delta(k;{\bf i})\}$ are the cluster variables belonging to a given initial seed in ${\mathbb C}[G^{u,v}]$ [Berenstein A., Fomin S., Zelevinsky A., Duke Math. J. 126 (2005), 1-52, math.RT/0305434]. In the case $G={\rm SL}_{r+1}({\mathbb C})$, $v=e$ and some special $u\in W$, we shall describe the generalized minors $\{\Delta(k;{\bf i})\}$ as summations of monomial realizations of certain Demazure crystals.